HYPERNUMBERS AND OTHER EXOTIC STUFF

Photo by Mateusz Dach

(1) MORE ON THE "ARITHMETICAL" SIDE

Tropical Introduction to Tropical - Diane Maclagan and Bernd Sturmfels http://www.cs.technion.ac.il/~janos/COURSES/238900-13/Tropical/MaclaganSturmfels.pdf https://en.wikipedia.org/wiki/Min-plus_matrix_multiplication https://en.m.wikipedia.org/wiki/Tropical_geometry#Algebra_background https://en.wikipedia.org/wiki/Amoeba_%28mathematics%29 https://www.youtube.com/watch?v=1_ZfvQ3o1Ac (friendly introduction) https://en.wikipedia.org/wiki/Log_semiring https://en.wikipedia.org/wiki/LogSumExp Tight spans, Isbell completions and semi-tropical modules - Simon Willerton https://arxiv.org/pdf/1302.4370.pdf (one half of the tropical semiring) Hyperfields for Tropical Geometry I. Hyperfields and dequantization - Oleg Viro https://arxiv.org/pdf/1006.3034.pdf (see section "6. Tropical addition of complex numbers") Supertropical quadratic forms II: Tropical trigonometry and applications - Zur Izhakian, Manfred Knebusch and Louis Rowen - https://www.researchgate.net/publication/ 326630264_Supertropical_Quadratic_forms_II_Tropical_Trigonometry_and_Applications Tropical geometry to analyse demand - Elizabeth Baldwin and Paul Klemperer http://elizabeth-baldwin.me.uk/papers/baldwin_klemperer_2014_tropical.pdf International Trade Theory and Exotic - Yoshinori Shiozawa https://link.springer.com/article/10.1007/s40844-015-0012-3

Arborescent numbers: higher operations and division trees - Henryk Trappmann http://eretrandre.org/rb/files/Trappmann2007_81.pdf Tetration Reference - Henryk Trappman and Andrew Robbins https://math.eretrandre.org/tetrationforum/attachment.php?aid=387 Open problems in tetration https://math.eretrandre.org/tetrationforum/showthread.php?tid=162 Applications of tetration https://math.stackexchange.com/questions/199862/what-is-the-geometric-physical-or-other-meaning- of-the-tetration

The family of arithmetics of Ruggero Maria Santilli http://www.santilli-foundation.org/docs/10.11648.j.ajmp.s.2015040501.14.pdf Isodual Theory of Antimatter with applications to Antigravity, Grand Unification and Cosmology https://www.amazon.com/Isodual-Theory-Antimatter-applications-Antigravity/dp/1402045174 (book) Studies on Santilli’s Isonumber Theory - Arun S. Muktibodh http://www.santilli-foundation.org/docs/pdf2.pdf Elements of Hadronic Mechanics III Experimental verifications - R.M.Santilli http://www.santilli-foundation.org/docs/elements-hadronic-mechanics-iii.compressed.pdf Initiating Santilli's Iso- to Triplex Number... - Nathan O. Schmidt and Reza Katebi http://vixra.org/pdf/1308.0051v2.pdf http://thunder-energies.com/ && http://www.santilli-foundation.org

The Development of Hyper-Dual Numbers for Exact Second-Derivative Calculations Jeffrey A. Fike and Juan J. Alonso http://adl.stanford.edu/hyperdual/Fike_AIAA-2011-886_slides.pdf

Saturation Arithmetics https://en.wikipedia.org/wiki/Saturation_arithmetic

Symmetric level index - https://en.wikipedia.org/wiki/Symmetric_level-index_arithmetic A Hybrid Number Representation Scheme Based on Symmetric Level-Index Arithmetic Xunyang Shen and Peter R. Turner - https://www.researchgate.net/publication/ 221142816_A_Hybrid_Number_Representation_Scheme_Based_on_Symmetric_Level- Index_Arithmetic

ZEA A zero-free exact arithmetic - Dominique Michelucci and Jean-Michel Moreau https://www.researchgate.net/publication/220991026_ZEA_-_A_zero-free_exact_arithmetic

Algebra of screws https://en.wikipedia.org/wiki/Screw_theory#Algebra_of_screws

On quantum state of numbers - Bernard Le Stum & Adolfo Quirós https://arxiv.org/pdf/1310.8143.pdf

Half-exponential function - https://en.wikipedia.org/wiki/Half-exponential_function exponential - https://en.wikipedia.org/wiki/Matrix_exponential Baker–Campbell–Hausdorff formula https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula

N-ary group https://en.wikipedia.org/wiki/N-ary_group

Circuits over sets of natural numbers https://en.wikipedia.org/wiki/Circuits_over_sets_of_natural_numbers The complexity of circuit evaluation over the natural numbers - Pierre McKenzie and Klaus Wagner http://www.iro.umontreal.ca/~mckenzie/Dagstuhl02.pdf

The Unwinding Number - Robert M. Corless and David J. Jeffrey https://faculty.e-ce.uth.gr/akritas/CE102/p28-corless.pdf

Generalized distributive law https://en.wikipedia.org/wiki/Generalized_distributive_law

A Noncommutative Version of the Natural Numbers - Tyler Foster https://arxiv.org/pdf/1003.2081.pdf

A new number system: Remainder numbers https://math.stackexchange.com/questions/2415896/a-new-number-system-remainder-numbers

Generalization of the unit interval - William M. Cornette https://projecteuclid.org/download/pdf_1/euclid.pjm/1102818012

Numeristics - Kevin Carmody - https://kevincarmody.com/math/numeristics.pdf

Construction, properties and applications of finite neofield - Anthony Donald Keedwell https://dml.cz/bitstream/handle/10338.dmlcz/119164/CommentatMathUnivCarolRetro_41-2000- 2_8.pdf

Quantity - https://en.wikipedia.org/wiki/Quantity_calculus http://ingvar.web03.cefit.se/wp-content/uploads/2016/02/physics6.pdf Metrological Thinking Needs the Notions of Parametric Quantities, Units, and Dimensions Ingvar Johansson

Solving Cubic Equations with Curly Roots - Dan Kalman and Maurice Burke https://www.jstor.org/stable/10.5951/mathteacher.108.5.0392?seq=1

The eightfold path to nonstandard analysis - Vieri Benci, Mauro Di Nasso and Marco Forti https://www.researchgate.net/profile/Vieri_Benci/publication/ 228753190_The_eightfold_path_to_nonstandard_analysis/links/0deec52e248b66edc1000000/The- eightfold-path-to-nonstandard-analysis.pdf

Radical of an integer - https://en.wikipedia.org/wiki/Radical_of_an_integer Integer root - https://en.wikipedia.org/wiki/Integer_square_root Quadratic residue - https://en.wikipedia.org/wiki/Quadratic_residue

Hypernumbers and Extrafunctions: Extending the Classical Calculus - Mark Burgin https://www.amazon.com/Hypernumbers-Extrafunctions-Extending-SpringerBriefs-Mathematics/dp/ 1441998748

Ordinal number - https://en.wikipedia.org/wiki/Ordinal_number Mex - https://en.wikipedia.org/wiki/Mex_(mathematics)

Parallel - https://en.wikipedia.org/wiki/Parallel_(operator)

Alternative models of the line in physics - D. K. Ross https://link.springer.com/article/10.1007/BF02213428 Can There Be an Alternative Mathematics, Really? - Jean Paul Van Bendegen https://link.springer.com/chapter/10.1007%2F0-387-24270-8_30 How Much Mathematics Is “Hardwired” If Any at All - Rafael Núñez https://cogsci.ucsd.edu/~nunez/COGS152_Readings/Nunez_ch3_MN.pdf

Fractions in transrational arithmetic - Jan A. Bregstra https://transmathematica.org/index.php/journal/article/view/19/23

Continuum between addition, multiplication and exponentiation https://math.stackexchange.com/questions/1269643/continuum-between-addition-multiplication-and- exponentiation

La quinta operación aritmética (The fifth arithmetical operation) New Numerical Methods: The Rational Mean (book) - Domingo Gomez Morin https://www.amazon.com/gp/product/1520717245/ref=dbs_a_def_rwt_hsch_vapi_tpbk_p1_i1 https://www.youtube.com/watch?v=6lORU03yuvY AULOS. LA OTRA LUZ. Music and Consonance. New musical scale not based on the Octave. https://www.youtube.com/watch?v=gbK_V_7ivDA https://domingogomezmorin.wordpress.com/

Setoid - https://en.wikipedia.org/wiki/Setoid

Some ternary quasigroups over small sets http://tamivox.org/dave/math/tern_quasi/index.html

The five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms - Kenneth A. Ribet – https://math.berkeley.edu/~ribet/trinity.pdf

Engel expansion - https://en.wikipedia.org/wiki/Engel_expansion Arithmetic Geometric Mean – https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, π, and the Ladies Diary Gert Almkvist and Bruce Berndt - https://link.springer.com/chapter/10.1007%2F978-3-319-32377-0_8

The total differential, the Cauchy-Riemann equations and the Elysian infinitesimals - Kerry Bemis Heinz mean - https://en.wikipedia.org/wiki/Heinz_mean Identric mean - https://en.wikipedia.org/wiki/Identric_mean Logarithmic mean - https://en.wikipedia.org/wiki/Logarithmic_mean

Knot sum - https://mathworld.wolfram.com/KnotSum.html

Hypertrascendental number - https://en.wikipedia.org/wiki/Hypertranscendental_number Infinite compositions of analytic functions https://en.wikipedia.org/wiki/Infinite_compositions_of_analytic_functions

Monus - https://en.wikipedia.org/wiki/Monus Racks and quandles - https://en.wikipedia.org/wiki/Racks_and_quandles Absorption law - https://en.wikipedia.org/wiki/Absorption_law

A quest for Exactness : machines, and geometry for tractional constructions of differential equations -Pietro Milici – https://tel.archives-ouvertes.fr/tel-01889365/document ( See section "7.3 Open problems and perspectives" )

Multiplicative calculus - https://en.wikipedia.org/wiki/Multiplicative_calculus Subderivative https://en.wikipedia.org/wiki/Subderivative Fractal derivative - https://en.wikipedia.org/wiki/Fractal_derivative

Alternative mathematical notation and its applications in calculus - Jakub Marian https://jakubmarian.com/data/bachelor_thesis.pdf

Mathematics Without Numbers Towards a Modal-Structural Interpretation - Geoffrey Hellman https://www.amazon.com/Mathematics-without-Numbers-Modal-Structural-Interpretation/dp/ 0198240341 Science Without Numbers A Defense of Nominalism - Hartry https://www.amazon.com/Science-without-Numbers-Hartry-Field/dp/0198777922

Los misterios de la fracción prohibida - Claudi Alsina and Carme Burgués The mysteries of the forbidden fraction (title translated) https://revistasuma.es/IMG/pdf/56/039-042.pdf Mediant - https://en.wikipedia.org/wiki/Mediant_(mathematics) https://en.wikipedia.org/wiki/Ford_circle https://en.wikipedia.org/wiki/Minkowski%27s_question-mark_function

Dialogue on n colored numbers - Armahedi Mahzar https://issuu.com/armahedimahzar/docs/dialogue_on_n-colored_nubers https://en.wikipedia.org/wiki/Additive_number_theory https://en.wikipedia.org/wiki/Zero-sum_problem https://en.wikipedia.org/wiki/Subset_sum_problem https://en.wikipedia.org/wiki/Restricted_sumset#Cauchy%E2%80%93Davenport_theorem

Some remarks on the pseudo- - Andrea markova https://www.sav.sk/journals/uploads/1203130414marko.pdf Pseudo-arithmetical operations as a basis for the general measure and integration theory - Pietro Benvenuti and Radko Mesiar - https://www.sciencedirect.com/science/article/pii/S0020025503002111

Polylogarithmic function https://en.wikipedia.org/wiki/Polylogarithmic_function

Hofstadter sequences - https://en.wikipedia.org/wiki/Hofstadter_sequence Mallows' Sequence - https://mathworld.wolfram.com/MallowsSequence.html

Negative Math: How Mathematical Rules Can Be Positively Bent (book) - Alberto A. Martínez https://www.amazon.com/Negative-Math-Mathematical-Rules-Positively-ebook/dp/B07DMVNZVP ( Algebra of quantities, history and variations of the algebra of signs )

Interval Arithmetic - https://en.wikipedia.org/wiki/Interval_arithmetic Theories of Interval Arithmetic Mathematical Foundations and Applications - Hend Dawood (book)https://www.academia.edu/1976964/Theories_of_Interval_Arithmetic_Mathematical_Foundatio ns_and_Applications

Graphs operations - https://en.wikipedia.org/wiki/Graph_operations

T.N.P - Tnp Socratis - https://groups.google.com/forum/#!forum/it.scienza.matematica

Summation 1+2+3+4+... https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF Umbral Calculus - https://en.wikipedia.org/wiki/Umbral_calculus

Progress Report on Hyper-operations (Zeration) https://math.eretrandre.org/tetrationforum/attachment.php?aid=251 Ackermann's Function and New Arithmetical Operations (zeration) http://www.rotarysaluzzo.it/Z_Vecchio_Sito/filePDF/Iperoperazioni%20(1).pdf Constantin A. Rubtsov and Giovanni F. Romerio

Constant problem - https://en.wikipedia.org/wiki/Constant_problem

Theory of holors (book) - and Domina Eberle Spencer https://www.amazon.com/Theory-Holors-Generalization-Moon-Spencer/dp/0521019001

J vocabulary- https://code.jsoftware.com/wiki/NuVoc Generalized inverse - https://en.wikipedia.org/wiki/Generalized_inverse Supermatrix - https://en.wikipedia.org/wiki/Supermatrix Hyperdeterminant - https://en.wikipedia.org/wiki/Hyperdeterminant

...some others can be found in https://en.wikipedia.org/wiki/List_of_types_of_numbers (2) MORE ON THE "NUMERAL" SIDE Lunar Arithmetic or Dismal Arithmetics - David Applegate, Marc LeBrun and N. J. A. Sloane https://cs.uwaterloo.ca/journals/JIS/VOL14/Sloane/carry2.pdf https://www.youtube.com/watch?v=cZkGeR9CWbk http://wwwinfo.deis.unical.it/~yaro/Numerals%20and%20Factorization.pdf

Balanced Ternary - https://en.wikipedia.org/wiki/Balanced_ternary https://pt.wikipedia.org/wiki/Tern%C3%A1rio_balanceado#/media/Ficheiro:Balanced_ternary.svg

Double-Base Number System for Multi- Multiplications Christophe Doche, David R. Kohel and Francesco Sica https://www.iacr.org/archive/eurocrypt2009/54790501/54790501.pdf

Skew binary number system - https://en.wikipedia.org/wiki/Skew_binary_number_system Two Skew-Binary Numeral Systems and One Application - Amr Elmasry and Jyrki Katajainen http://cphstl.dk/Paper/TOCS-2011/journal.pdf

Zero Displacement Ternary Number System : the most economical way of representing numbers - Fernando Guilherme Silvano Lobo Pimentel https://www.researchgate.net/publication/ 258241283_Zero_Displacement_Ternary_Number_System_the_most_economical_way_of_representin g_numbers

Quote Notation - Eric C. R. Hehner and R. N. S. Horspool http://www.cs.toronto.edu/~hehner/ratno.pdf https://en.wikipedia.org/wiki/Quote_notation

Beyond the Complexes: Toward a based number system - J. Köplinger, J. A. Shuster https://www.cs.du.edu/~petr/milehigh/2013/Koeplinger.pdf

Linear Numeral System - Ian Mackie - http://www.ianmackie.com/papers/linns.pdf

New approach could sink floating point computation, John Leroy Gustafson https://www.nextplatform.com/2019/07/08/new-approach-could-sink-floating-point-computation/ https://en.wikipedia.org/wiki/Double-precision_floating-point_format

The residue logarithmic number system: Theory and implementation - Mark G.Arnold https://www.researchgate.net/publication/ 4156476_The_residue_logarithmic_number_system_Theory_and_implementation

A Low-Power Two-Digit Multi-dimensional Logarithmic Number System Filterbank Architecture for a Digital Hearing Aid -- Roberto Muscedere, Vassil Dimitrov, Graham Jullien and William Miller https://www.researchgate.net/publication/26532063_A_Low-Power_Two-Digit_Multi- dimensional_Logarithmic_Number_System_Filterbank_Architecture_for_a_Digital_Hearing_Aid

Methodology of numerical computations with infinities and infinitesimals - Yaroslav D. Sergeyev http://www.theinfinitycomputer.com/The_second_paper_to_read_(Lagrange_Lecture).pdf https://www.numericalinfinities.com/ Hetero Base Arithmetic Operation - Raghavendra Lingayya (????) http://www.numbersystem.org/hetero-base-arithmetic-operations.html

Zot-Binary: a new numbering system with an application on big-integer multiplication - Shahram Jahani and Azman Samsudin - http://www.jatit.org/volumes/Vol48No1/5Vol48No1.pdf

Universal Script http://www.dscript.org/ Matthew DeBlock ( Uscript is universal logographic language based on math and physics )

Decimal Fractions - https://en.wikipedia.org/wiki/Simon_Stevin#Decimal_fractions

A Number System with Continuous Valued Digits and Modulo Arithmetic - Aryan Saèd, Majid Ahmadi and Graham A. Jullien - https://www.academia.edu/13000520/

A_number_system_with_continuous_valued_digits_and_modulo_arithmetic Hereditary Base notation - https://en.wikipedia.org/wiki/Goodstein%27s_theorem#Hereditary_base-n_notation

New Arithmetic Algorithms for Hereditarily Binary natural numbers - Paul Tarau https://www.cse.unt.edu/~tarau/research/2014/HBinX.pdf

Predicting Improper Fractional Base Integer Characteristics - Billy Dorminy http://educ.jmu.edu/~lucassk/Papers/DorminyFracBase.pdf

Horus Eye Fractions - https://en.wikipedia.org/wiki/Eye_of_Horus#Mathematics Finger Binary - https://en.wikipedia.org/wiki/Finger_binary Nemeth braille - https://en.wikipedia.org/wiki/Nemeth_Braille Bibi-binary -- https://en.wikipedia.org/wiki/Bibi-binary

Quater-imaginary base - https://en.wikipedia.org/wiki/Quater-imaginary_base

QUANTUM-LANGUAGE-PARSE-SYNTAX-GRAMMAR https://en.wikipedia.org/wiki/David_Wynn_Miller#Miller's_description_of_his_work_and_views https://github.com/lismore/MathematicalInterfaceForLanguage/blob/master/README.md https://dwmlc.com/

The Denormal Logarithmic Number System - Mark G. Arnold Sylvain Collange https://www.researchgate.net/publication/262371524_The_Denormal_Logarithmic_Number_System

The generalized golden proportions, a new theory of real numbers, and ternary mirror-symmetrical arithmetic - Alexey Stakhov - http://fs.unm.edu/SN/TheGeneralizedGolden.pdf

Construction of Algorithms for Parallel Addition - Jan Legersky and Milena Svobodová https://jan.legersky.cz/talks/ConstructionParAddAlg_WorkshopOnAutomaticSequences.pdf

On-line algorithms for multiplication and division in real and complex numeration systems - Marta Brzicová, Christiane Frougny, Edita Pelantová and Milena Svobodová https://arxiv.org/abs/1610.08309v5 Computing with Exact Real Numbers in a Radix-r System - Alexander Kaganovsky https://www.researchgate.net/publication/ 220368828_Computing_with_Exact_Real_Numbers_in_a_Radix-r_System

A variant of Ostrowski numeration - Emmanuel Cabanillas https://arxiv.org/pdf/1904.01874v2.pdf https://oeis.org/wiki/LCM_numeral_system https://oeis.org/wiki/Factorial_numeral_system https://en.wikipedia.org/wiki/Factorial_number_system https://oeis.org/wiki/Primorial_numeral_system http://www.thefullwiki.org/Combinadic

Octomatics number system - http://octomatics.org/

Sandpiles - Luis David Garcia-Puente https://www.youtube.com/watch?v=1MtEUErz7Gg http://people.reed.edu/~davidp/

Super omega - https://en.wikipedia.org/wiki/Chaitin%27s_constant#Super_Omega

Esolang – https://esolangs.org/wiki/Main_Page https://en.wikipedia.org/wiki/Approximate_number_system https://en.wikipedia.org/wiki/Numerical_cognition https://en.wikipedia.org/wiki/Number_sense_in_animals Together with dyscalculia, ageometresia, dysgraphia, financial illiteracy https://en.wikipedia.org/wiki/Innumeracy_%28book%29 https://www.andnextcomesl.com/2019/10/hypernumeracy.html

Location arithmetic - https://en.wikipedia.org/wiki/Location_arithmetic

Yupana - https://en.wikipedia.org/wiki/Yupana

...some can be found in the following wikipedia links : https://en.wikipedia.org/wiki/List_of_numeral_systems https://en.wikipedia.org/wiki/Category:Non-standard_positional_numeral_systems https://en.wikipedia.org/wiki/Non-standard_positional_numeral_systems

Real Computation https://en.wikipedia.org/wiki/Real_computation Hypercomputation https://en.wikipedia.org/wiki/Hypercomputation

Unconventional computing (list) https://en.wikipedia.org/wiki/Unconventional_computing

(3) MORE ON THE "GEOMETRICAL" SIDE Cubic Pythagoras – Luis Teia (pythagoras with instead of ) https://wonderfulengineering.com/pythagoras-theorem-has-been-upgraded-to-3d-and-now-requires-a- 120-page-proof/ https://www.youtube.com/channel/UCcoPpQFHQsv6pDzMgVI_pRw/videos? view=0&sort=dd&shelf_id=0

A Mathematical Theory of Origami Constructions and Numbers - Roger C. Alperin https://arxiv.org/pdf/math/9912039v1.pdf Teoría de Galois tras el Origami - Alberto Garcia Diaz https://riull.ull.es/xmlui/bitstream/handle/915/5795/Teoria%20de%20Galois%20tras%20el%20origami. %20Por%20que%20el%20origami%20resuelve%20los%20problemas%20geometricos%20clasicos %20de%20la%20Antigua%20Grecia..pdf?sequence=1&isAllowed=y Origami-Constructible Numbers - James King https://www.cs.mcgill.ca/~jking/papers/origami.pdf Origami and Partial Differential Equations - Bernard Dacorogna, Paolo Marcellini and Emanuele Paolini

Theory of 3D complex space and of 3D space, applications and experimental validation techniques - L.T. Abobda https://www.researchgate.net/publication/ 301627462_Theory_of_3D_complex_space_and_complex_number_of_3D_space_applications_and_ex perimental_validation_techniques

Hoop Algebras - Roger Beresford https://library.wolfram.com/infocenter/MathSource/6198/ https://demonstrations.wolfram.com/author.html?author=Roger+Beresford https://library.wolfram.com/infocenter/search/?search_results=1&search_person_id=4705 https://mathworld.wolfram.com/AlgebraicLoop.html https://groupprops.subwiki.org/wiki/Moufang_loop (Ruth Moufang) Smarandache Loops - W. B. Vasantha Kandasamy http://fs.unm.edu/Vasantha-Book4.pdf

Truly hypecomplex numbers : Unification of numbers and vectors - Redouane Bouhennache https://arxiv.org/pdf/1409.2757.pdf

On a novel 3D system - Shlomo Jacobi https://arxiv.org/pdf/1509.01459.pdf

Hypercomplex number in three dimensional spaces - Abdelkarim Assoul https://www.researchgate.net/publication/ 308969073_Hypercomplex_number_in_three_dimensional_spaces_hal-01686021

Introduction to the General Trigonometry in Euclidian 2D-space - Claude Ziad Bayeh http://www.wseas.us/e-library/transactions/mathematics/2012/53-882.pdf

Solving Quadratic Equations - Peter Michael Jack https://archive.org/details/q2wp01

A System of Three-Dimensional complex variables - E. Dale Martin https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19880004569.pdf

Rational trigonometry - Norman J. Wildberger https://www.youtube.com/user/njwildberger https://en.wikipedia.org/wiki/Rational_trigonometry

Generalizaciones de los números: de la aritmética a las variedades diferenciables - Fernando Etayo Gordejuela https://repositorio.unican.es/xmlui/bitstream/handle/ 10902/13817/2016GacRSocMatEspGeneralization.pdf?sequence=1&isAllowed=y

Sinc function - https://en.wikipedia.org/wiki/Sinc_function

Surprises and pitfalls arising from (pseudo)symmetry - P. H. Zwart, R. W. Grosse-Kunstleve, A. A. Lebedev, G. N. Murshudov and P. D. Adams https://journals.iucr.org/d/issues/2008/01/00/ba5111/ba5111.pdf

Ensemble de nombres - Taladris, Silk78, Seirios, Telchar, Tigerfou and Médiat

Fractals arithmétiques - Jean-Pierre Reveilles http://numerisation.univ-irem.fr/ST/IST93018/IST93018.pdf List of fractals by Hausdorff dimension https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension Fractal wheel https://ksr-ugc.imgix.net/assets/004/987/498/d1d3926f15a17d6194a07825630d3424_original.gif? ixlib=rb-2.1.0&w=680&fit=max&v=1448600022&auto=format&gif- q=50&q=92&s=9851a96b94a4aaab1fdf587ccd3e5647 An Intrinsically Three-Dimensional Fractal -- M. Fernández-Guasti https://www.researchgate.net/publication/267132753_An_Intrinsically_Three-Dimensional_Fractal

List of Coordinate Systems https://en.wikipedia.org/wiki/Category:Coordinate_systems https://www.gbv.de/dms/goettingen/198419775.pdf

On the Extension of Complex Numbers - Nicholas Gauguin Houghton-Larsen https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.304.5052&rep=rep1&type=pdf

Trilinear Coordinates - https://mathworld.wolfram.com/TrilinearCoordinates.html Barycentric Coordinates - https://mathworld.wolfram.com/BarycentricCoordinates.html Synergetics Coordinates - https://mathworld.wolfram.com/SynergeticsCoordinates.html Special Isocubics in the Triangle Plane - Jean-Pierre Ehrmann and Bernard Gibert https://bernard-gibert.pagesperso-orange.fr/files/Resources/SITP.pdf Transformation of trilinear and quadriplanar to and from cartesian coordinates -John B Mertie http://www.minsocam.org/ammin/AM49/AM49_926.pdf

An introduction to the perplex number system - Jerry Chandler https://core.ac.uk/download/pdf/81127362.pdf

Bucky Number Mandelbrot - Clifford J. Nelson https://library.wolfram.com/infocenter/MathSource/428/

A New and Very Long Proof of the Pythagoras Theorem - Kaushik Basu http://kaushikbasu.org/Pythagoras%206.pdf

Blog about the 3d complex numbers and other related stuff - http://3dcomplexnumbers.net/

The Literal Calculus of Viete and Descartes - I. G. Bashmakova and G. S. Smirnova https://historiamatecuaciones.files.wordpress.com/2012/07/the-literal-calculus-of-viete-and- descartes.pdf

P-adics numbers The p-adic integers - Brian Courthoute, Pablo Guzman and Antoine Ronk http://math.uni.lu/eml/projects/reports/P-adics.pdf A first introduction to p-adic numbers - David A. Madore http://www.madore.org/~david/math/padics.pdf

Polynumbers, Norms, Metrics, and Polyingles - R R Aidagulov and M V Shamolin https://www.researchgate.net/publication/270597014_Polynumbers_Norms_Metrics_and_Polyingles Finsler Spaces, Bingles, Polyingles, and Their Symmetry Groups - R. R. Aidagulov and Maxim V. Shamolin https://www.researchgate.net/publication/ 270597384_Finsler_Spaces_Bingles_Polyingles_and_Their_Symmetry_Groups

Web "3d Math Secrets" (????) https://www.3dmathsecrets.com/breakthrough https://www.skills31teams.com/about-the-professor https://www.csop.global/about-us

The non-equality between curve and the straight line - Walter Meyer http://curiosidadesmatematicas.cl/wordpress/aclaracion/ https://curiosidadesgeometricas.blogspot.com/2017/ http://curiosidadesmatematicas.cl/wordpress/espanol-matematicas/espanol-analisis-de-la-no-igualdad- de-la-curva-y-la-recta-extracto/ https://www.youtube.com/user/Curiosidadesgeo/

Isotropic line - https://en.wikipedia.org/wiki/Isotropic_line

Matemática Discreta Isodimensional - http://www.isodimensional.org/

Generalization of 3D Mandelbrot and Julia sets - Cheng Jin and Tan Jian-rong https://www.deepdyve.com/lp/springer-journals/generalization-of-3d-mandelbrot-and-julia-sets- GXA2OHcHRA

Understanding & Using "nuReal numbers" - John A. Shuster

Proportion functions in three dimensions - Claudi Alsina and Walter Benz https://link.springer.com/article/10.1007/BF01836452 Misbehaved lines https://i.stack.imgur.com/kYCs0.png https://static.scientificamerican.com/blogs/cache/file/AEE64282-EF71-4F2C- AA9557A9188E1C2F_agenda.jpghttps://xorshammer.files.wordpress.com/2010/03/sheaf2_line.png https://www.modelrailforum.com/forums/uploads/1439922327/gallery_15688_407_150853.jpg

An algorithm for multiplication of trigintaduonions – Alexandr Cariow and Galina Cariowa https://pdfs.semanticscholar.org/2a77/5a4f39ba0a0d1ceb34b3e0a1c2223117d680.pdf

Circular and Hyperbolic , Octonions, and Sedenions - Kevin Carmody https://www.sciencedirect.com/science/article/abs/pii/0096300388901336

Geometry of Generalized Complex Numbers - Anthony Harkin and Joseph B. Harkin https://www.researchgate.net/publication/265769569_Geometry_of_Generalized_Complex_Numbers

Musean hypernumbers - http://www.house-of-horus.de/hypernumbers.html https://en.wikipedia.org/w/index.php?title=Hypernumber&oldid=78200756 https://plus.wikimonde.com/wiki/Hypernombre Elliptic complex numbers with dual multiplication - John Shuster and Jens Koplinger http://www.jenskoeplinger.com/P/PaperShusterKoepl_WSpace.pdf Doubly nilpotent numbers in the 2D plane - John Shuster and Jens Koplinger http://www.jenskoeplinger.com/P/PaperShusterKoepl-PQSpace.pdf

Fractal dimension and Wada measure revisited : no straightforward relationships in NDDS - Pranas Ziaukas and Minvydas Ragulskis https://nonlinear.fmf.ktu.lt/Papers/ND_2017_v2.pdf https://en.wikipedia.org/wiki/Lakes_of_Wada

Foundations of transcomplex numbers An extension of the complex number system to four dimensions - Perez Ernesto

Three Gears are Possible – Numberphile - https://www.youtube.com/watch?v=5Mf0JpTI_gg Bashing Geometry with Complex Numbers, Evan Chen https://web.evanchen.cc/handouts/cmplx/en-cmplx.pdf

Trigonometry of a tetrahedron - https://en.wikipedia.org/wiki/Trigonometry_of_a_tetrahedron

Complex Numbers The Higher Dimensional Forms 2nd Edition - Dennis Morris https://www.amazon.com/gp/product/1508677492/ref=dbs_a_def_rwt_bibl_vppi_i16

Solid Geometry with Problems and Applications - H. E. Slaught and N. J. Lennes https://www.gutenberg.org/files/29807/29807-pdf.pdf

Introduction to the circular number line - Dharmendra Kumar Yadav https://www.researchgate.net/publication/ 301552425_INTRODUCTION_OF_A_CIRCULAR_NUMBER_LINE A new approach to ordering complex numbers - Dharmendra Kumar Yadav https://www.researchgate.net/publication/ 267465398_A_new_approach_to_ordering_complex_numbers

Transfinity A Source Book - Wolfgang Mückenheim https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf THE ANT LIST V 4.0 - Sergio https://groups.google.com/g/sci.math/c/WN-gBszU8ko

M.E. Irizarry-Gelpí https://meirizarrygelpi.github.io/posts/maths/beyond-complex/index.html https://godoc.org/github.com/meirizarrygelpi/rational

N-dimensional complex numbers - http://www.alenspage.net/ComplexNumbers.htm

The vector algebra war: a historical perspective - James M. Chappell, Azhar Iqbal, John G. Hartnett, and Derek Abbott https://arxiv.org/pdf/1509.00501.pdf

Polysign Numbers - Tim Golden http://www.bandtechnology.com/PolySigned/index.html Notas Sobre Polisignos Y Objetos Tertiarios - Kujonai https://vixra.org/pdf/2002.0570v1.pdf

Paravector - https://en.wikipedia.org/wiki/Paravector Multivector - https://en.wikipedia.org/wiki/Multivector

Tau manifesto - https://tauday.com/tau-manifesto https://hexnet.org/files/documents/tau-manifesto.pdf

Hypercomplex Numbers in Geometry and Physics (Scientific Journal) http://hypercomplex.xpsweb.com/section.php?lang=en&genre=3

Using Chinese Dumbass Notation to Find Algebraic Identities Daniel - Liu Daniel Liu https://www.academia.edu/11576181/ Using_Chinese_Dumbass_Notation_to_Find_Algebraic_Identities

Nonions of James Joseph Sylvester A Synopsis of Linear Associative Algebra - James Byrnie Shaw https://babel.hathitrust.org/cgi/pt?id=coo.31924062544949&view=1up&seq=97

New Calculus - John Gabriel https://www.youtube.com/channel/UClBbBVLs3M-d3dNgU4Vop_A/videos http://thenewcalculus.weebly.com/

A complex and Triplex framework for encoding the riemannian dual space-time equipped with order parameters fields - N. O. Schmidt https://www.researchgate.net/publication/ 236735724_A_complex_and_triplex_framework_for_encoding_the_Riemannian_dual_space- time_topology_equipped_with_order_parameter_fields

The simple complex numbers , Jaroslaw Zalesny https://arxiv.org/abs/0802.0312 https://en.wikipedia.org/wiki/Double_Fourier_sphere_method

A Possible Solution of Trisection Problem - Siavash H. Sohrab http://www.wseas.us/e-library/conferences/2012/CambridgeUSA/MATHCC/MATHCC-44.pdf

A Three Dimensional Coordinate System for Complex Numbers - Greg Ehmka http://gregehmka.com/math-ebook

Pseudo-vector - https://en.wikipedia.org/wiki/Pseudovector Pseudo-scalar - https://en.wikipedia.org/wiki/Pseudoscalar Pseudotensor - https://en.wikipedia.org/wiki/Pseudotensor

Iconic Arithmetic - William Bricken - http://iconicmath.com/ https://archive.org/details/iconicarithmetic01will/mode/2up

Approach on coordinate, coordinate an their usage in true 3dgis Gang Liao, Qingyuan Li, Xu Chen and Jiarong Zheng https://www.researchgate.net/publication/ 242605764_APPROACH_ON_AREA_COORDINATE_VOLUME_COORDINATE_AND_THEIR_S AGE_IN_TRUE_3DGIS Areal Co-ordinate Methods in - Tom Lovering https://bmos.ukmt.org.uk/home/areals.pdf

Semi-Complex Analysis & - F. Antonuccio - https://arxiv.org/pdf/gr-qc/9311032.pdf

Hex Grid Geometry for Game Developers - Herman Tulleken http://gamelogic.co.za/downloads/HexMath2.pdf

Cognitive-Theoretic Model of the Universe ( CTMU ) Christopher Langan http://hology.org/ Non-well-founded theory - https://en.wikipedia.org/wiki/Non-well-founded_set_theory

Mathematical surprises in 3d (some observations and open problems in r3) http://claudialsina.com/sorpresas-matematicas-en-3d

"Quaternions - Redundancy + Efficiency = Ternions" - Ulrich Mutze http://www.ulrichmutze.de/articles/05-53.pdf

OMIC's N-nion's site - anonymous author - http://asyncbrain.baf.cz/m/nt/index.htm The trinion Fourier transform of color images - Dawit Assefa, Lalu Mansinha, Kristy F. Tiampo, Henning Rasmussen and Kenzu Abdella - https://www.academia.edu/3835064/The_trinion_Fourier_transform_of_color_images Three-Dimensional Wind Profile Prediction with Trinion-Valued Adaptive Algorithms - Zhi Wen Liu, Wei Liu and You Gen Xu - https://www.researchgate.net/publication/278048724_Three- Dimensional_Wind_Profile_Prediction_with_Trinion-Valued_Adaptive_Algorithms Vectors, Cyclic Submodules and Projective Spaces Linked with Ternions - Hans Havlicek and Metod Saniga - https://www.researchgate.net/publication/ 1737480_Vectors_Cyclic_Submodules_and_Projective_Spaces_Linked_with_Ternions

Wasan Geometry and Division by Zero Calculus - Hiroshi Okumura and Saburou Saitoh https://www.researchgate.net/publication/ 329210266_Wasan_Geometry_and_Division_by_Zero_Calculus http://okmr.yamatoblog.net/

Ternary numbers and algebras - Alexey Dubrovski and Guennadi Volkov https://arxiv.org/pdf/hep-th/0608073.pdf

On Unconventional Division by Zero - Jakub Czajko http://www.worldscientificnews.com/wp-content/uploads/2018/04/WSN-99-2018-133-147.pdf

The sextonions and E - Landsberg, J. M., & Manivel, L. https://arxiv.org/pdf/math/0402157.pdf Sextonions, Zorn Matrices, and e71/ 2 https://arxiv.org/abs/1506.04604v1 https://en.wikipedia.org/wiki/E7%C2%BD Sextonions and the magic square - Bruce W. Westbury https://arxiv.org/abs/math/0411428

The Great Pi Conspiracy - Mark and Scott Wollum https://omnithought.org/great-pi-conspiracy/2584

Mathematics of Archimedes Plutonium https://groups.google.com/forum/?hl=en#!forum/plutonium-atom-universe

Andre Joyce http://untilheaven.tripod.com/transfinite_mathematics_made_easy.htm http://untilheaven.tripod.com/andre_joyce_s_coined_words.htm

Quasic blog - L. Edgar Otto - https://pesla.blogspot.com/

Spiritual Mathematics: Introduction to the Circular Number System – John Dunne-Brady https://books.google.cl/books? id=dDPgAgAAQBAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=fal se

Nova processes - Ted Gress - https://www.twilightraven.net/ http://vixra.org/pdf/1804.0337v1.pdf

Tetryonics – https://tetryonics.com/

Crank Dot Net - List of bizarre mathematics Erik Max Francis – http://www.crank.net/maths.html

Where is the frontier between "mathematics" and "pseudo-mathematics"? https://en.wikipedia.org/wiki/Pseudomathematics Pseudo-mathematics VS Proto-mathematics Can exist "dissident" mathematicians in a similar way of "dissident" scientists? https://www.academia.edu/37679452/Jean_de_Climont_-_The_worldwide_list- of_dissident_scienticts_1-500_-_Part_1.pdf

(5) LISTS OF LISTS OF OPEN PROBLEMS

Darpa 23 Maths Problems https://compmath.wordpress.com/about/10-the-big-picture-darpas-23-challenge-questions/ Problems of the Wolfram Project https://www.wolframscience.com/openproblems/NKSOpenProblems.pdf http://mathworld.wolfram.com/UnsolvedProblems.html Open problems in Mathematics - John Forbes Nash Jr and Michael Rassias http://www.mthrassias.com/data/uploads/bfm3a978-3-319-32162-22f1.pdf

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